Group #135708 — subgroups

Subgroups — #135708

Order 512.

216 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#7842	C₂	1	1	yes	maximal, characteristic
2	#7195	C₂	1	2	yes	maximal
3	#7148	C₂	1	2	yes	maximal
4	#7145	C₂	1	2	yes	maximal
5	#2162	C₂²	1	4	yes	—
6	#1744	C₄	1	1	yes	characteristic
7	C₄×#97	C₄	1	1	yes	characteristic
8	C₄²×C₈	C₂²	1	1	yes	characteristic
9	C₂×#322	C₄	1	1	yes	characteristic
10	C₂×#322	C₄	1	1	yes	characteristic
11	C₂×#322	C₂²	1	1	yes	characteristic
12	C₂×#322	C₂²	1	1	yes	characteristic
13	C₂×#97	—	2	2	no	—
14	C₂×C₄×C₈	—	2	2	no	—
15	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
21	C₄³	C₂×C₄	1	1	yes	characteristic
22	C₄⋊C₁₆	—	2	4	no	—
23	C₂²⋊C₁₆	—	2	4	no	—
24	C₄×C₁₆	—	2	4	no	—
25	#336	C₈	1	4	yes	—
26	#322	—	2	2	no	—
27	#322	—	2	2	no	—
28	#322	—	2	2	no	—
29	#322	—	2	2	no	—
30	C₂²×C₈	—	2	1	no	—
31	C₂²×C₈	—	2	1	no	—
32	C₂²×C₈	C₂×C₈	1	2	yes	—
33	C₂²×C₈	C₂×C₈	1	2	yes	—
34	C₂²×C₈	M₁₆	1	2	yes	—
35	C₂²×C₈	#41	1	2	yes	—
36	C₂×C₄²	—	2	1	no	—
37	C₂×C₄²	—	2	1	no	—
38	C₂×C₄²	C₂×C₈	1	2	yes	—
39	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₁₆	—	4	4	no	—
41	#97	—	4	4	no	—
42	C₄×C₈	—	2	1	no	—
43	C₄×C₈	—	2	1	no	—
44	C₄×C₈	—	2	1	no	—
45	C₄×C₈	—	2	1	no	—
46	C₄×C₈	—	2	1	no	—
47	C₄×C₈	—	2	1	no	—
48	C₄×C₈	—	2	1	no	—
49	C₄×C₈	—	2	1	no	—
50	C₄×C₈	—	4	4	no	—
51	C₄×C₈	M₁₆	1	2	yes	—
52	C₄×C₈	M₁₆	1	2	yes	—
53	C₄×C₈	#41	1	2	yes	—
54	C₄×C₈	#41	1	2	yes	—
55	C₂²×C₄	—	2	2	no	—
56	C₂²×C₄	#117	1	1	yes	characteristic
57	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
58	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
59	C₂×C₈	—	2	4	no	—
60	C₂×C₈	—	2	2	no	—
61	C₂×C₈	—	2	2	no	—
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	2	2	no	—
64	C₂×C₈	—	2	2	no	—
65	C₂×C₈	—	2	4	no	—
66	C₂×C₈	—	2	4	no	—
67	C₂×C₈	—	4	2	no	—
68	C₂×C₈	—	4	2	no	—
69	C₂×C₈	—	4	2	no	—
70	C₂×C₈	#126	1	4	yes	—
71	C₄²	—	2	4	no	—
72	C₄²	—	2	4	no	—
73	C₄²	—	4	2	no	—
74	C₄²	#117	1	1	yes	characteristic
75	C₄²	#117	1	1	yes	characteristic
76	C₄²	C₂×M₁₆	1	1	yes	characteristic
77	C₄²	C₂×M₁₆	1	1	yes	characteristic
78	C₁₆	—	8	4	no	—
79	C₂³	#431	1	1	yes	characteristic
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	2	4	no	—
83	C₂×C₄	—	4	2	no	—
84	C₂×C₄	—	4	2	no	—
85	C₂×C₄	#383	1	1	yes	characteristic
86	C₂×C₄	#431	1	1	yes	characteristic
87	C₂×C₄	#431	1	1	yes	characteristic
88	C₂×C₄	#432	1	1	yes	characteristic
89	C₂×C₄	#432	1	1	yes	characteristic
90	C₂×C₄	#432	1	1	yes	characteristic
91	C₂×C₄	#504	1	4	yes	—
92	C₈	—	2	4	no	—
93	C₈	—	2	4	no	—
94	C₈	—	4	2	no	—
95	C₈	—	4	2	no	—
96	C₈	—	4	2	no	—
97	C₈	—	4	2	no	—
98	C₂²	—	2	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	#1839	1	1	yes	characteristic
101	C₂²	#2164	1	2	yes	—
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	—	4	4	no	—
105	C₄	#1841	1	2	yes	—
106	C₄	#2164	1	2	yes	—
107	C₄	#2164	1	2	yes	—
108	C₄	#2165	1	2	yes	—
109	C₂	—	2	2	no	—
110	C₂	#8801	1	1	yes	characteristic
111	C₂	#12895	1	1	yes	characteristic
112	C₂	#12896	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#7842

C₂

yes

maximal, characteristic

#7195

C₂

yes

maximal

#7148

C₂

yes

maximal

#7145

C₂

yes

maximal

#2162

C₂²

yes

—

#1744

C₄

yes

characteristic

C₄×#97

C₄

yes

characteristic

C₄²×C₈

C₂²

yes

characteristic

C₂×#322

C₄

yes

characteristic

C₂×#322

C₄

yes

characteristic

C₂×#322

C₂²

yes

characteristic

C₂×#322

C₂²

yes

characteristic

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄³

C₂×C₄

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

#336

C₈

yes

—

—

—

—

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×C₈

yes

—

C₂²×C₈

C₂×C₈

yes

—

C₂²×C₈

M₁₆

yes

—

C₂²×C₈

#41

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×C₈

yes

—

C₂×C₄²

C₂²×C₄

yes

characteristic

C₂×C₁₆

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

#41

yes

—

C₄×C₈

#41

yes

—

C₂²×C₄

—

C₂²×C₄

#117

yes

characteristic

C₂²×C₄

C₂²×C₈

yes

characteristic

C₂²×C₄

C₂×M₁₆

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#126

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

yes

C₄²

yes

C₄²

C₂×M₁₆

yes

characteristic

C₄²

C₂×M₁₆

yes

characteristic

C₁₆

—

C₂³

#431

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

—

100

C₂²

#1839

yes

characteristic

101

C₂²

#2164

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

#1841

yes

—

106

C₄

#2164

yes

—

107

C₄

#2164

yes

—

108

C₄

#2165

yes

—

109

C₂

—

110

C₂

#8801

yes

characteristic

111

C₂

#12895

yes

characteristic

112

C₂

#12896

yes

characteristic

	subgroup	G/H
center	C₄²	#117
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1839
prime radical	C₂³	#431
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₄²

#117

commutator

C₂²×C₄

C₂²×C₈

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₄² ◁ C₂×#322 ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#7842	2
#7195	2
#7148	2
#7145	2

subgroup

index

Subgroups — #135708

Conjugacy classes of subgroups

Maximal subgroups